90
Views
10
CrossRef citations to date
0
Altmetric
Original Articles

A heteroclinic network in mode interaction with symmetry

, &
Pages 359-396 | Received 30 Nov 2009, Accepted 01 Jul 2010, Published online: 15 Oct 2010

References

  • Golubitsky , M , Stewart , IN and Schaeffer , D . 1988 . “ Singularities and Groups in Bifurcation Theory. Volume 2 ” . In Appl. Math. Sci. , Vol. 69 , New York : Springer-Verlag .
  • Golubitsky , M and Stewart , IN . 2002 . “ The Symmetry Perspective ” . In Progress in Mathematics , Vol. 200 , Basel : Birkhäuser .
  • Aguiar , MAD , Castro , SBSD and Labouriau , IS . 2005 . Dynamics near a heteroclinic network . Nonlinearity , 18 : 391 – 414 .
  • Aguiar , MAD , Labouriau , IS and Rodrigues , AAP . 2010 . Switching near a network . Dynamical Sys. , 25 : 75 – 95 .
  • Aguiar , MAD and Castro , SBSD . Chaotic switching in a two-person game , Working paper 305 FEP, 2008 (to appear in Physica D)
  • Ashwin , P , Rucklidge , AM and Sturman , R . 2004 . Two state intermittency near a symmetric interaction of saddle-noda and Hopf bifurcations: A case study from dynamo theory . Physica D , 194 ( 1–2 ) : 30 – 48 .
  • Armbruster , D , Stone , E and Kirk , V . 2003 . Noisy heteroclinic networks . Chaos , 13 ( 1 ) : 71 – 79 .
  • Postlethwaite , CM and Dawes , JHP . 2005 . Regular and irregular cycling near a heteroclinic network . Nonlinearity , 18 : 1477 – 1509 .
  • Kirk , V , Lane , E , Postlethwaite , CM , Rucklidge , AM and Silber , M . 2010 . A mechanism for switching near a heteroclinic network . Dyn. Sys. , 25 ( 3 ) : 323 – 349 .
  • Kirk , V and Rucklidge , AM . 2008 . The effect of symmetry-breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit . Dyn. Sys. , 23 ( 1 ) : 43 – 74 .
  • Küppers , G and Lortz , D . 1969 . Transition from laminar convection to thermal turbulence in a rotatin fluid layer . J. Fluid Mech. , 35 : 609 – 620 .
  • Busse , FH and Heikes , KE . 1980 . Convection in a rotating layer: A sample case of turbulence . Science , 208 : 173 – 175 .
  • Bodenschatz , E , Pesch , W and Ahlers , G . 2000 . Recent developments in Rayleigh-Bénard convection . Ann. Rev. Fluid Mech. , 32 : 709 – 778 .
  • Proctor , MRE and Jones , CA . 1988 . The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance . J. Fluid Mech. , 188 : 301 – 355 .
  • Porter , J and Knobloch , E . 2001 . New type of complex dynamics in the 1:2 spatial resonance . Physica D , 159 : 125 – 154 .
  • Mercader , I , Prat , J and Knobloch , E . 2002 . Robust heteroclinic cycles in two-dimensional Rayleigh–Bénard convection without Boussinesq symmetry . Int. J. Bif. Chaos , 12 : 281 – 308 .
  • Mizushima , J and Fujimura , K . 1992 . Higher harmonic resonance of two-dimensional disturbances in Rayleigh–Bénard convection . J. Fluid Mech. , 234 : 651 – 667 .
  • Porter , J and Knobloch , E . 2000 . Complex dynamics in the 1:3 spatial resonance . Physica D , 143 : 138 – 168 .
  • Podvigina , OM and Ashwin , PB . 2007 . The mode interaction and heteroclinic networks in Boussinesq convection . Physica D , 234 : 23 – 48 .
  • Podvigina , O . 2010 . Stability of rolls in rotating magnetoconvection in a layer with no-slip electrically insulating horizontal boundaries . Phys. Rev. E , 81 : 056322
  • Golubitsky , M , Swift , JW and Knobloch , E . 1984 . Symmetries and pattern selection in Rayleigh–Bénard convection . Physica D , 10 : 249 – 276 .
  • Getling , AV . 1998 . Rayleigh–Bénard Convection: Structures and Dynamics , Singapore : World Scientific Publishing .
  • Anosov , DV , Aranson , SKh , Arnold , VI , Bronshtein , IU , Grines , VZ and Il'yashenko , YuS . 1988 . “ Ordinary Differential Equations and Smooth Dynamical Systems ” . In Dynamical Systems I, Vol. I of Encyclopædia of Mathematical Sciences , Berlin : Springer .
  • Deng , B . 1989 . The Sil'nikov problem, exponential expansion, strong λ-lemma, C 1-linearization and homoclinic bifurcation . J. Diff. Eqns. , 79 : 189 – 231 .
  • Samovol , VS . 1979 . Linearisation of systems of ordinary differential equations in a neighbourhood of invariant toroidal manifolds . Proc. Moscow Math. Soc. , 38 : 187 – 219 .
  • Sternberg , S . 1957 . Local contractions and a theorem of Poincaré . Amer. J. Maths. , 79–4 : 802 – 824 .
  • Kirk , V and Silber , M . 1994 . A competition between heteroclinic cycles . Nonlinearity , 7 : 1605 – 1621 .
  • Brannath , W . 1994 . Heteroclinic networks on the tetrahedron . Nonlinearity , 7 : 1367 – 1384 .
  • Krupa , M and Melbourne , I . 1995 . Asymptotic stability of heteroclinic cycles in systems with symmetry . Ergodic Theory and Dynam. Sys , 15 : 121 – 147 .
  • Krupa , M and Melbourne , I . 2004 . Asymptotic stability of heteroclinic cycles in systems with symmetry II . Proc. Roy. Soc. Edinburgh , 134A : 1177 – 1197 .
  • Melbourne , I . 1991 . An example of a non-asymptotically stable attractor . Nonlinearity , 4 : 835 – 844 .
  • Homburg , AJ and Knobloch , J . 2009 . Switching homoclinic networks , Amsterdam : preprint . (to appear in Dynamical Systems)
  • Hu , Y , Ecke , RE and Ahler , G . 1997 . Convection under rotation for Prandtl number near 1: Linear stability, wave-number selection and pattern dynamics . Phys. Rev. E , 55 : 6928 – 6949 .
  • Fujimura , K and Yamada , S . 2008 . Hexagons and triangles in the Rayleigh-Bénard problem: quintic-order equation on a hexagonal lattice . Proc. R. Soc. A , 464 : 2721 – 2739 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.